QAMA: The Calculator That Makes You Better At Math

From a post by Alex Knapp on Forbes blog:

QAMA […] is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math skills.  It does this in a deceptively simple way.

The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise answer.

Quick – what’s the square root of 2? What do you mean you don’t have a calculator? Well, you can start guessing, right? So let’s work this through. You know you have an upper bound – it has to be less than 1.5, because 1.5 x 1.5 is 2.25.  And it has to be more than 1, because 1 x 1 is just 1. But 2.25 is pretty close, right? So what if you guess 1.4? Well, then you’d be pretty close. 1.4 x 1.4 is 1.96, and the square root of 2 is about 1.414.

But did you notice something? Without your calculator, you had to estimate. In order to estimate, you had to think about and engage with the math behind exponents and square roots. Which means, hopefully, that you came out of that first paragraph with a bit better understanding of math.

That’s the theory behind QAMA, which is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math skills.  It does this in a deceptively simple way.

The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise answer.

How close is close? Well, that depends on the calculation. If you put in 5×6, you have to estimate 30 – the calculator expects you to know your multiplication tables. For exponent problems, if you have an integer – say something like 23^2, the tolerance is such that you still have to be pretty close, but there’s a wider berth for say, 23^2.1, because non-integer exponents are a tougher problem.

Developing the calculator to have these different tolerances for different types of calculations was the key challenge for QAMA inventor Ilan Samson.

Read the original post in full.


Rethinking maths for the 21st century

From Research News on the University of  Cambridge website:

An exciting new Maths Education Programme is being launched by the University of Cambridge which aims to provide innovative, rich and stimulating materials to help support and inspire teachers and students of advanced post-16 mathematics.

The Project will receive £2.8 million from the Department for Education over the initial three years of the five year project, with a review after three years.

It will be led by Professor Martin Hyland, head of the Department of Pure Mathematics and Mathematical Statistics, and Lynne McClure, director of NRICH, part of

the University’s Millennium Mathematics Project.

The programme will seek to reconsider and rethink how changes in our understanding of maths impact on the mathematics which is studied at school level. The past few decades have seen advances in our understanding of core mathematics, major developments in areas such as probability and the emergence of new disciplines, including mathematical biology.

It will provide rich resources for advanced post-16 mathematics which will augment and support current teaching, be published online and be freely accessible to all. The emphasis will be on simple underlying mathematical ideas, helping students to explore connections between different areas of mathematics, and supporting the development of key mathematical skills and clarity of thought. The impetus for the programme comes from a belief in the importance of dialogue between schools, higher education and research.

Building on the University of Cambridge’s long history of working with schools, for instance through the Millennium Mathematics Project, researchers will consult widely with teachers during the development of the programme. While individual students will also be able to work through the resources independently, the project will provide extensive teacher support material to encourage classroom use. In addition, the programme will include professional development summer schools for teachers. The University of Cambridge programme will also work closely with other organisations supporting advanced post-16 mathematics.

It is anticipated that pilot versions of material will begin to be published next summer, with development continuing over the following two years.

Professor Hyland says: “We are very grateful for this opportunity to share thinking about the major themes in mathematics with teachers. One of the key aims of the project is to provide material to support inspirational and committed teachers in exploring the subject beyond curriculum boundaries, leading to a richer educational experience for all.”

Cambridge University ‘to set maths A-levels’

From  ‘s article in The Telegraph:

Leading mathematicians are to script new syllabuses and exam questions as part of radical reforms being introduced to drive up education standards.

Revised qualifications will feature an emphasis on key disciplines such as trigonometry and probability, “demanding” questions will be set to stretch the brightest pupils and lesson materials will be available online.

The move is designed to address major concerns over a sharp decline in teenagers’ maths skills – leaving hundreds of thousands of young people unfit for the demands of higher education.

Cambridge warned that even the most talented students did not have “sufficient mastery of basic mathematics” and existing A-levels were too “superficial”.

Academics including Sir Tim Gowers, who won the prestigious Fields Medal for mathematics in 1998, will be involved in the project, although it could lead to a significant delay in the introduction of new sixth-form exams. […]

A source close to Michael Gove said: “It is vital we raise standards, raise ambition and get people who really understand subjects back in charge.

“It is incredibly exciting that some of the best mathematicians in the world want to fix A-level maths. This will spread understanding of teaching the deep problem-solving skills that are so vital to universities and businesses, and give many more pupils an advanced education.”

Cambridge’s Department of Pure Mathematics has submitted a report to the Department for Education outlining how new-style maths A-levels should be structured.

It claims that changes are needed because “the majority of the talented students which Cambridge is able to recruit do not have sufficient mastery of basic mathematics to enable them to confidently engage with anything other than routine problems”.

“Existing A-level curriculums treat topics superficially and the UK has lost the tradition of teaching school mathematics coherently and in depth,” it adds. “The effect on Cambridge is acute.”

The document, by Prof Martin Hyland, head of the department, suggests focusing A-levels around a series of “key mathematical ideas”. This is likely to include complex numbers, trigonometry, combinatorics, probability and centres of mass.

In a key change, it recommends creating “graded sets of problems” for bright teenagers. A major part of assessments will be addressed at all students, but Cambridge is proposing a “range of demanding questions to challenge the most able”.

Academics are pledging to “exploit the potential of the web” by making maths materials available online and creating a newly-constructed website for teachers’ feedback.

Mathematicians from other universities will be asked for their input into the new A-level, which will extensively trialled in schools.

But the move is likely to lead to an overall delay

to the introduction of new-style qualifications in the subject, with Cambridge suggesting they could take five years to develop.

The Department for Education originally suggested it wanted new A-levels to be taught for the first time in 2014, although the Cambridge plan would rule out major changes until 2017 at the earliest.

Read the full article.


Tony Gardiner: A mathematician’s view of the current education scene in the UK

The current turmoil facing mathematics within the UK educational scene – from primary to postgraduate – is unprecedented in my experience. At the same time, the formal institutions and agencies on which we all depend have never been weaker. At undergraduate level the issues are mostly UK-wide; but they are being interpreted and tackled independently by individual universities, and by different groupings (Russell Group, University Alliance, 1994, Million+). The school-level agenda is complicated by the fact that the UK has four different education systems, one of which is far bigger, and more turbulent, than the other three. In England, the last three administrations (from 1979) have adopted policies that replace traditional collegiality with competition and ‘market forces’: they may speak the language of devolution; but implementation often concentrates effective control at the centre. Changes in Wales, Scotland and Northern Ireland are often less visible to most of us – with Scotland having its own strong traditions, while Wales and Northern Ireland are more influenced by what happens in England.

Continue reading

The ownership of A-levels

From TES, from an article Gove under fire, by William Stewart, 19 October 2012:

“By far the most important thing we are doing on A levels

is getting university academics back in the driving seat instead of the Department for Education,” the source [very close to education secretary Michael Gove] said. […]

[Mr Gove] wanted government to “step back”, allowing universities to take “real and committed” ownership of new A levels, giving the qualifications their endorsement so that they, rather than exam boards, “drive the system”.

But in an official response to the plans, seen by TES, Universities UK states: “We do not think it would be advisable or operationally feasible for the sector to take on the ‘ownership of the exams’, particularly in terms of formally endorsing all A levels as currently  proposed.”

[UUK] argues that because A levels are a national qualification, “ultimate responsibility and accountability” for them should remain with the government. […]

Ministers believe there is a split over A levels between academics and the universities they work for, which represents a “huge problem”.

“Almost all academics want linear A levels, but universities are not run by academics and admin offices have totally different views, partly because of the cursed focus on ‘access’ which has poisoned intelligent discussion of (the) real problem, which is too many rubbish schools,” the source close to Mr Gove said.

Read the whole article.

Alan Milburn’s report on social mobility

Independent Reviewer’s report on Higher Education (called in media “Alan Milburn’s Report“) is published today. Here is one of the bits related to mathematics:

Equalising skills
Universities can do more to ensure that students have the essential skills they require to complete their degrees. Applicants from

disadvantaged backgrounds are less likely to have developed certain skills, such as essay writing. Clearly some university courses will rightly require a high level of prior knowledge, for example, some science subjects require students to have a high degree of mathematical knowledge on day one in order to succeed. Other courses will have far fewer direct constraints. […]

Universities should consider what support they can provide to help particular groups of under-represented students succeed in completing their studies. In some cases, this will require assessing what skills universities require students to have in advance and which ones they can develop after admission.

British Academy: “Society Counts”

British Academy published today report Society Counts: Quantitative Skills in the Social Sciences and Humanities (link to full text). A quote:

Statistical literacy for UK graduates

20. The British Academy has frequently emphasised the need for well-rounded graduates, equipped with core skills, if the UK is to retain its status in research and higher education.
21. These core skills start with quantitative methods. The skills standardly deployed, for example, in the natural sciences and engineering are no longer synonymous with or restricted to particular subjects; these skills are now relevant and necessary well beyond traditional science, technology, engineering and mathematics (STEM ) subjects. The changes required to develop these skills in graduates is relevant across the university curriculum. We must therefore seek to apply some of the methods and thinking

that are being used to bring about curriculum change through the STEM initiative.

(Link to full text of the Report)

A Bacc is coming?

From Mail Online, By Ben Spencer:

Education secretary Michael Gove is […] said to be developing an Advanced Baccalaureate which would see students studying a mixture of A-level subjects, writing a 5,000-word essay and undertaking voluntary work. […]

If his proposals are enacted it would mean the entire exam system for secondary schools will have been replaced in the space of thee years.

The new baccalaureate system would require A-level students to study ‘contrasting’ subjects to give them a broad education, The Times reported last night.

A candidate who chose A levels in maths, further maths and physics, for example, would be expected to pick a humanity, such as history or French, as a fourth subject. 

Mr Gove also wants shorter and more open-ended questions in exams.

One option he is apparently considering is to limit the A-Bacc to teenagers who choose at least two A levels from a list of subjects specified by Russell Group universities – maths, further maths, English literature, physics, biology, chemistry, geography, history, and modern and classical


Read more:

Mathematics and What It Means to Be Human, Part 1

An article by Michele Osherow and Manil Suri  in The Chronicle of Higher Education. From an introduction:

In May 2009, Michele Osherow, an English professor at the University of Maryland-Baltimore County and dramaturg at the Folger Theatre, in Washington, invited her colleague Manil Suri, a mathematician at the university, to act as

mathematics consultant for the Folger’s production of Tom Stoppard‘s ArcadiaThe play explores the relationship between past and present through the characters’ intellectual pursuits, poetic and mathematical. That led to a series of “show and tell” sessions explaining the mathematics behind the play to both cast members and audiences. In the fall of 2011, the two professors decided to take their collaboration to the classroom and jointly teach a freshman seminar, “Mathematics and What It Means to be Human.” Here is the first of a three-part series on how the experiment played out.

Read the full article.

New exams without trials?

From The Guardian:

Education Guardian has learned that Ofqual, the exams regulator, has quietly abandoned a promise to ensure that all major exam reforms are piloted in advance. This means that the next big set of changes – the much-discussed introduction of English Baccalaureate Certificates (EBCs) to replace GCSEs, initially in

English, maths and science, from 2015 – are likely to go ahead without any conventional pre-trials.

Read the full article.